Abstract
We present a robust algorithm starting from 1D or 2D discrete noised data to approximately invert the heat equation, which is an ill-conditioned problem. Relative contributions of the coherent structure and the noise in different frequency bands of the available data are different. We propose to solve the inversion problem separately in different frequency bands by methods similar to the Tikhonov regularization. This separation is achieved by using spline wavelet packets. The solutions are derived as linear combinations of those wavelet packets.
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References
Averbuch A, Zheludev V (2009) Spline-based deconvolution. Signal Process 89(9):1782–1797
Averbuch A, Zheludev V, Khazanovsky M (2011) Deconvolution by matching pursuit using spline wavelet packets dictionaries. Appl Comput Harmon Anal 31(1):98–124
Averbuch A, Zheludev V, Neittaanmäki P, Koren J (2010) Block based deconvolution algorithm using spline wavelet packets. J Math Imaging Vis 38(3):197–225
Coifman RR, Wickerhauser MV (1992) Entropy-based algorithms for best basis selection. IEEE Trans Inf Theory 38(2):713–718
Donoho D, Johnstone I (1994) Ideal spatial adaptation via wavelet shrinkage. Biometrika 81(3):425–455
Fourier J (1822) Theorie analytique de la chaleur. Firmin Didot, Paris. Reissued by Cambridge University Press, Cambridge, 2009
Tikhonov AN (1963) Solution of incorrectly formulated problems and the regularization method. Sov Math Dokl 4:1035–1038
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Averbuch, A., Neittaanmäki, P., Zheludev, V. (2013). Inversion of the Heat Equation by a Block Based Algorithm Using Spline Wavelet Packets. In: Repin, S., Tiihonen, T., Tuovinen, T. (eds) Numerical Methods for Differential Equations, Optimization, and Technological Problems. Computational Methods in Applied Sciences, vol 27. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5288-7_12
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DOI: https://doi.org/10.1007/978-94-007-5288-7_12
Publisher Name: Springer, Dordrecht
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